Results 1 
9 of
9
GROUP RECONSTRUCTION SYSTEMS
 ELA
, 2011
"... We consider classes of reconstruction systems (RS’s) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS’s), that are associated with representations of finite groups G. These GRS’s generalize frames with high degree of symmetry, such as harmonic or geom ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We consider classes of reconstruction systems (RS’s) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS’s), that are associated with representations of finite groups G. These GRS’s generalize frames with high degree of symmetry, such as harmonic or geometrically uniform ones. Their canonical dual and canonical Parseval are shown to be GRS’s. We establish simple conditions for oneerasure robustness. Projective GRS’s, that can be viewed as fusion frames, are also considered. We characterize the Gram matrix of a GRS in terms of block group matrices. Unitary equivalences and unitary symmetries of RS’s are studied. The relation between the irreducibility of the representation and the tightness of the GRS is established. Taking into account these results, we consider the construction of Parseval, projective and oneerasure robust GRS’s.
HARMONIC RECONSTRUCTION SYSTEMS
 ELA
, 2013
"... This paper considers group reconstruction systems (GRS’s), for finite dimensional real or complex Hilbert spaces H, that are associated with unitary representations of finite abelian groups. The relation between these GRS’s and the generalized Fourier matrix is established. A special type of Parsev ..."
Abstract
 Add to MetaCart
(Show Context)
This paper considers group reconstruction systems (GRS’s), for finite dimensional real or complex Hilbert spaces H, that are associated with unitary representations of finite abelian groups. The relation between these GRS’s and the generalized Fourier matrix is established. A special type of Parseval GRS, called harmonic reconstruction system (HRS), is defined. It is shown that there exist HRS’s that present maximal robustness to erasures given characterizations of certain families.
Author manuscript, published in "SAMPTA'09, Marseille: France (2009)" Gradient descent of the frame potential
, 2010
"... Unit norm tight frames provide Parsevallike decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential ..."
Abstract
 Add to MetaCart
(Show Context)
Unit norm tight frames provide Parsevallike decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential. We consider this minimization problem from a numerical perspective. In particular, we discuss how by descending the gradient of the frame potential, one, under certain conditions, is guaranteed to produce a sequence of unit norm frames which converge to a unit norm tight frame at a geometric rate. This makes the gradient descent of the frame potential a viable method for numerically constructing unit norm tight frames. 1.
Minimizers of the fusion frame potential
, 2008
"... In this paper we study the fusion frame potential, that is a generalization of the BenedettoFickus (vectorial) frame potential to the finitedimensional fusion frame setting. Local and global minimizers of this potential are studied, when we restrict it to a suitable set of fusion frames. These min ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper we study the fusion frame potential, that is a generalization of the BenedettoFickus (vectorial) frame potential to the finitedimensional fusion frame setting. Local and global minimizers of this potential are studied, when we restrict it to a suitable set of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. Thus, we exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the socalled HornKlyachko’s compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, varying the sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different